# How to interpret fixed effects model when the fixed effects uniquely identifies each observation?

Say I have a dataset that is uniquely identified by country and year. Then, I run a fixed effects regression with country and year fixed effects. How do I interpret the result of the regression coefficient? Surely I cannot say that it is the average correlation of X on Y controlling for countries and years, because each country and each year uniquely identify my observations! And yet commands such as areg in STATA claim that they are able to do just that. Perhaps I am misunderstanding something. Somebody please help.

• country $\times$ year is not the same as country AND year FEs. Commented Jan 26, 2022 at 21:56
• I am interested in the latter. I do understand that the former would also uniquely identify all my observations, but doesn't the latter as well? Commented Jan 26, 2022 at 23:04
• No, compare what you get with webuse pig, clear reg weight i.id i.week reg weight i.id##i.week  Commented Jan 26, 2022 at 23:12
• I see you're trying to point out that if I added country AND year FEs, then there would still be degrees of freedom available to estimate standard errors. but how am I to interpret the coefficient, especially when both country and year FEs are present as well? the effect of X on Y, controlling for year and country (i.e. if I were to add country FEs, then I would interpret it as "the effect of X on Y within country Z", but with BOTH year and country FEs, I could not say "the effect of X on Y within country Z and year T", because Z and T uniquely identify X and Y...). Commented Jan 27, 2022 at 1:14

$$E[y_i \vert x_i,c_i,t]=\beta \times x_i + \gamma_{ci}+ \eta_t$$
You can think of the $$\gamma_{ci}$$ as a country-specific intercept that does not vary across time. It captures all the time-invariant factors for each country (like longitude). $$\eta_t$$ is a time effect that is constant across countries but is not time-invariant (like oil prices). These two effects will shift the y-x curve up or down, leaving the slope with respect to $$x$$ alone.
The interpretation of $$\beta$$ is the change in the expected value of $$y_i$$ associated with a 1 unit increase in $$x_i$$. In this case, it is the same for all countries and time periods. It's the marginal effect of $$x$$ adjusting for differences in time-invariant characteristics across countries and for time effects that impact all countries in the same way.